. Geometry and Locus of Complex Numbers In this section let us study the geometrical interpretation of complex number z in complex plane and the locus of z in Cartesian form. Example . Given the complex number z , represent the complex numbers z iz iz , and in one Argand plane.
Show that these complex numbers form the vertices of an isosceles right triangle. Given that z . Therefore, iz iz Let A B , , and C be z z iz iz respectively. AB iz BC iz iz CA iz Since AB BC CA and AB BC , D ABC is an isosceles right triangle.
Definition . (circle) A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point in that plane is always a constant. The fixed point is the centre and the constant distant is the radius of the circle. Equation of Complex Form of a Circle The locus of z that satisfies the equation z where z is a fixed complex number and r is a fixed positive real number consists of all points z whose distance from z is r .
Therefore z is the complex form of the equation of a circle. (see Fig. . ) (i) z represents the points interior of the circle.
(ii) z represents the points exterior of the circle. Illustration . , represents a circle centre at the origin with radius r units. - - -